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Table 7.4
Discernibility
functions related to 9 kinds of
reducts
Reduct
Discernibility function(s)
Exact/approximate
F
L
ʲ
L
Exact
F
U
ʲ
U
Exact
(
F
B
F
L
F
U
ʲ
B
ʲ
,
ʲ
∧
)
Approximate
(
F
P
F
L
ʲ
P
ʲ
,
)
Approximate
(
F
UN
ʲ
F
U
ʲ
UN
,
)
Approximate
F
L
F
U
ʲ
LU
ʲ
∧
Exact
(
F
P
ʲ
∧
F
UN
F
L
F
U
ʲ
PUN
,
ʲ
∧
)
Approximate
ʲ
ʲ
∧
F
UN
F
L
F
L
F
U
LUN
(
,
ʲ
∧
ʲ
)
Approximate
ʲ
(
F
B
ʲ
∧
F
P
F
L
F
U
BUN
ʲ
,
ʲ
∧
ʲ
)
Approximate
The obtained discernibility functions are shown in Table
7.4
. In the case of approx-
imate discernibility functions, the first function in the parenthesis characterizes
the necessary condition of the preservation and the second function characterizes
the sufficient condition. The discernibility functions related to LU-reducts, LUN-
reducts and BUN-reducts can be obtained by taking the conjunctions of discernibil-
ity functions related to L-reducts, U-reducts, B-reducts and UN-reducts. Note that
F
B
ʲ
∧
F
UN
F
L
F
U
F
U
F
L
F
U
ʲ
. This is why we have
F
L
F
U
ʲ
as
the discernibility function characterizing a sufficient condition for the preservation
of BUN-reducts.
ʲ
=
(
ʲ
∧
ʲ
)
∧
ʲ
=
ʲ
∧
ʲ
∧
Example 9
Remember the decision table
D =
(
U
,
C
∪{
d
}
,
{
V
a
}
)
in Table
7.3
. Let an
admissible error rate be
ʲ
=
0
.
39. In Table
7.5
, we show the decision table with three
0
.
39
0
.
39
0
.
39
generalized decision functions
39.
Now let us enumerate reducts as prime implicants of discernibility functions. First
let us discuss L-, U- and LU-reducts with
ʻ
,
˅
,
(˅
\
ʻ)
with respect to
C
and
ʲ
=
0
.
C
C
C
ʲ
=
0
.
39. The discernibility matrix of the
decision table is shown as below.
P
1
P
2
P
3
P
4
P
5
P
1
∅{
c
3
,
c
4
}
{
c
1
}
{
c
1
,
c
2
}{
c
2
,
c
3
}
P
2
{
c
3
,
c
4
}
∅{
c
1
,
c
3
,
c
4
}
C
{
c
2
,
c
4
}
P
3
{
c
1
}{
c
1
,
c
3
,
c
4
}
∅ {
c
2
}{
c
1
,
c
2
,
c
3
}
P
4
{
c
1
,
c
2
}
C
{
c
2
}
∅
{
c
1
,
c
3
}
P
5
{
c
2
,
c
3
}{
c
2
,
c
4
}{
c
1
,
c
2
,
c
3
}{
c
1
,
c
3
}
∅
Table 7.5
The decision table in Table
7.3
with the generalized decision functions
c
1
ʻ
C
0
.
39
0
.
39
c
2
c
3
c
4
d
:(b,m,g)
˅
(˅
\
ʻ)
C
C
P
1
Good
Good
Bad
Good
(0,2,9)
{g}
{g}
∅
P
2
Good
Good
Good
Bad
(0,19,1)
{m}
{m}
∅
P
3
Bad
Good
Bad
Good
(1,1,2)
∅
{g}
{g}
P
4
Bad
Bad
Bad
Good
(0,1,1)
∅
{m,g}
{m,g}
P
5
Good
Bad
Good
Good
(1,1,1)
∅
∅
∅
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